Question

Two balls with the same mass and initial velocity are projected at different angles in such a way that the maximum height reached by the first ball is 8 times higher than that of the second ball. $ T_1 $ and $ T_2 $ are the total flying times of the first and second ball, respectively, then the ratio of $ T_1 $ and $ T_2 $ is:

• A. \( 2 : 1 \)
• B. \( \sqrt{2} : 1 \)
• C. \( 4 : 1 \)
• D. \( 2\sqrt{2} : 1 \)

Hint:

In projectile motion, the total time of flight and maximum height are related to the angle of projection. For a given initial velocity, the height and time of flight depend on \( \sin^2 \theta \) and \( \sin \theta \), respectively.

Answer 1

The Correct Option is D

To determine the ratio of the total flying times \( T_1 \) and \( T_2 \) for two balls projected with the same initial velocity but at different angles, given that the maximum height of the first ball is 8 times that of the second, follow these steps:

1. Recall the formulas for maximum height (\( H \)) and time of flight (\( T \)) in projectile motion. Maximum height is given by:

\(H = \frac{v_i^2 \sin^2 \theta}{2g}\)

1. where:
   • \(v_i\) is the initial velocity.
   • \(\theta\) is the projection angle.
   • \(g\) is the acceleration due to gravity.
2. The total time of flight (\( T \)) is calculated as:

\(T = \frac{2v_i \sin \theta}{g}\)

1. The problem states that the height of the first ball (\( H_1 \)) is 8 times the height of the second ball (\( H_2 \)):

\(\frac{H_1}{H_2} = 8\)

1. Substitute the height formula into the ratio:

\(\frac{\frac{v_i^2 \sin^2 \theta_1}{2g}}{\frac{v_i^2 \sin^2 \theta_2}{2g}} = 8\)

1. Simplify the expression:

\(\frac{\sin^2 \theta_1}{\sin^2 \theta_2} = 8\)

1. The objective is to find the ratio of the total times of flight:

\(\frac{T_1}{T_2} = \frac{\frac{2v_i \sin \theta_1}{g}}{\frac{2v_i \sin \theta_2}{g}} = \frac{\sin \theta_1}{\sin \theta_2}\)

1. From step 3, we have the relationship:

\(\frac{\sin^2 \theta_1}{\sin^2 \theta_2} = 8\)

1. Take the square root of both sides:

\(\frac{\sin \theta_1}{\sin \theta_2} = \sqrt{8} = 2\sqrt{2}\)

Thus, the ratio of the total flying times \( T_1 \) to \( T_2 \) is \( 2\sqrt{2} : 1 \).